5 ms·

This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or us

by samth 4y ago

This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or useless, but is independent of ZFC.

SemanticStrengh 4y ago
independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? Also, I am aware there are non-contrived statements that are can't be proven in ZFC such as https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#:~:text=A%20statement%20is%20independent%20of,from%20the%20axioms%20of%20ZFC https://en.wikipedia.org/wiki/List_of_statements_independent.... I'm only claiming that the incompleteness theorems only find contrived ones, and that for the non-contrived ones humanity has found, it is NOT that they are mathematically unproveable, it is that ZFC is not enough.
- JadeNB 4y ago
  > independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? There are plenty of people who find ZFC + ¬CH a reasonable axiomatic system. EDIT: I misread samth's statement (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581), which I think is proveable in ZFC; I thought their statement was ¬CH.
  - SemanticStrengh 4y ago
    It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?
    - JadeNB 4y ago
      I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).
      - SemanticStrengh 4y ago
        I am making the distinction between natural statements and contrived statements. I claim godel only find contrived statements. However there exist non-contrived, natural statements that are unprovable in ZFC. My question is, what set of additional axioms are needed to have a maximal coverage/provability of those independent statements, while introducing the least possible axioms. And yes indeed ZFC + ¬CH seems like a good first step.
        CaptainNegative 4y ago
        There are a large number of barriers to formalizing your question, such as (i) the definition of natural, (ii) the definition of "maximal" when comparing countably infinite sets, and (iii) the fact that asking about the provability of X in vanilla ZFC is still arguably interesting in the system ZFC+X (see: reverse mathematics), so it's not clear what adding it as an axiom does towards the "largeness" of the set of provable natural statements.
        SemanticStrengh 4y ago
        About formalising (i) I guess it's about accepting everything that isn't a meta-token, I don't remember godel incompleteness theorems sufficently to be precise about it, anyway the useful list would be the wikipedia list of independent statements since most of them are not contrived (although very abstract). So yes my question can be specialized as to how to achieve maximum coverage of the wikipedia page. Otherwise, great comment and it's been a while since I hadn't heard the term reverse mathematics!
  - samth 4y ago
    My statement was slightly misphrased, it was intended to be just a statement of CH.
- samth 4y ago
  There are a variety of stronger axiomatic systems (typically ZFC + some other axioms). Some of them prove the continuum hypothesis (which is what I stated), others prove the negation. There's no way to decide which one is "right", though, just like there's no way to tell if the Axiom of Choice is "right". Your second paragraph is confused. Given an axiom system like ZFC, there are (a) statements that can be proved true or false using it, (b) statements that can't be proved that are true in a particular model, (c) statements that can't be proved that are false in that model. It's set (b) that the incompleteness theorem tells us must exist. The theorem doesn't "find" statements, it proves that (b) exists by constructing a particular one, which is necessarily meta because it applies to every formal system. However, we do not have access to a model which tells us the answers for things like the CH. So all you can do is decide on the axiom scheme you like, and then some things are provable. You can always add more axioms, like CH, but you can add their negation instead, if you want. So there's no sense in which there's really a right answer for CH but we haven't found it yet.
- pfortuny 4y ago
  Well, Gödel set out to prove incompleteness. The first statement that comes to mind is “there is an unprovable sentence” but that is too general. What Gödel did was find out that “this statement is unprovable” is “definable” in Peano’s system (once it has been encoded), and that’s it. Nothing especially contrived there. Less so for a professional logician.

by samth 4y ago

This is not true. For example, the statement "there exists a bijection between the real numbers and the powerset of the integers" is not contrived or meta or useless, but is independent of ZFC.

SemanticStrengh 4y ago

independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? Also, I am aware there are non-contrived statements that are can't be proven in ZFC such as https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#:~:text=A%20statement%20is%20independent%20of,from%20the%20axioms%20of%20ZFC https://en.wikipedia.org/wiki/List_of_statements_independent.... I'm only claiming that the incompleteness theorems only find contrived ones, and that for the non-contrived ones humanity has found, it is NOT that they are mathematically unproveable, it is that ZFC is not enough.

JadeNB 4y ago
> independently of ZFC, can your statement be mathematically proven in a reasonable alternative axiomatic system? There are plenty of people who find ZFC + ¬CH a reasonable axiomatic system. EDIT: I misread samth's statement (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581), which I think is proveable in ZFC; I thought their statement was ¬CH.
- SemanticStrengh 4y ago
  It seems its not enough ? https://mathoverflow.net/questions/273861/independence-over-zfc-ch https://mathoverflow.net/questions/273861/independence-over-... (Although its CH not the negation of it) what additional axioms be needed there?
  - JadeNB 4y ago
    I misread the initial statement as "there exist sets intermediate in size between the real numbers and the integers", which may have been what samth meant (https://news.ycombinator.com/item?id=31424581 https://news.ycombinator.com/item?id=31424581). That is ¬CH, so ZFC + ¬CH can obviously prove it. If you meant that ZFC + ¬CH is not enough to prove interesting mathematics, then it surely is, since ZFC itself is already enough to prove interesting mathematics. If you meant that ZFC + ¬CH (or ZFC + CH) is not enough to decide all "natural" statements … well, I agree with that, but it seemed that you were arguing against, not for, that position (https://news.ycombinator.com/item?id=31423948 https://news.ycombinator.com/item?id=31423948).
    - SemanticStrengh 4y ago
      I am making the distinction between natural statements and contrived statements. I claim godel only find contrived statements. However there exist non-contrived, natural statements that are unprovable in ZFC. My question is, what set of additional axioms are needed to have a maximal coverage/provability of those independent statements, while introducing the least possible axioms. And yes indeed ZFC + ¬CH seems like a good first step.
      - CaptainNegative 4y ago
        There are a large number of barriers to formalizing your question, such as (i) the definition of natural, (ii) the definition of "maximal" when comparing countably infinite sets, and (iii) the fact that asking about the provability of X in vanilla ZFC is still arguably interesting in the system ZFC+X (see: reverse mathematics), so it's not clear what adding it as an axiom does towards the "largeness" of the set of provable natural statements.
        SemanticStrengh 4y ago
        About formalising (i) I guess it's about accepting everything that isn't a meta-token, I don't remember godel incompleteness theorems sufficently to be precise about it, anyway the useful list would be the wikipedia list of independent statements since most of them are not contrived (although very abstract). So yes my question can be specialized as to how to achieve maximum coverage of the wikipedia page. Otherwise, great comment and it's been a while since I hadn't heard the term reverse mathematics!
- samth 4y ago
  My statement was slightly misphrased, it was intended to be just a statement of CH.
samth 4y ago
There are a variety of stronger axiomatic systems (typically ZFC + some other axioms). Some of them prove the continuum hypothesis (which is what I stated), others prove the negation. There's no way to decide which one is "right", though, just like there's no way to tell if the Axiom of Choice is "right". Your second paragraph is confused. Given an axiom system like ZFC, there are (a) statements that can be proved true or false using it, (b) statements that can't be proved that are true in a particular model, (c) statements that can't be proved that are false in that model. It's set (b) that the incompleteness theorem tells us must exist. The theorem doesn't "find" statements, it proves that (b) exists by constructing a particular one, which is necessarily meta because it applies to every formal system. However, we do not have access to a model which tells us the answers for things like the CH. So all you can do is decide on the axiom scheme you like, and then some things are provable. You can always add more axioms, like CH, but you can add their negation instead, if you want. So there's no sense in which there's really a right answer for CH but we haven't found it yet.
pfortuny 4y ago
Well, Gödel set out to prove incompleteness. The first statement that comes to mind is “there is an unprovable sentence” but that is too general. What Gödel did was find out that “this statement is unprovable” is “definable” in Peano’s system (once it has been encoded), and that’s it. Nothing especially contrived there. Less so for a professional logician.